Related papers: Randomness below complete theories of arithmetic
In this short note we show that every connected reductive simply-connected algebraic group of rank $>1$ over the complex numbers has infinitely many pairs of irreducible representations which are not related by an automorphism of the…
Spatially embedded networks are important in several disciplines. The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known. Here we present new results for the two-point degree…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in…
We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to generalized hypergeometric functions…
This paper deals with (finite or infinite) sequences of arbitrary independent events in some probability space. We find sharp lower bounds for the probability of a union of such events when the sum of their probabilities is given. The…
Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In…
A real \alpha is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to \alpha. It is known that the randomness of an r.e. real \alpha can be characterized in…
We analyze the threshold network model in which a pair of vertices with random weights are connected by an edge when the summation of the weights exceeds a threshold. We prove some convergence theorems and central limit theorems on the…
Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…
We show that every strongly jump-traceable set obeys every benign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
Preferential attachment graphs are random graphs designed to mimic properties of typical real world networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already…
We extend the notions of "$R_\infty$-property" and "full (extended) Reidemeister spectrum" to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small…
We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…
Do complexity classes have many-one complete sets if and only if they have Turing-complete sets? We prove that there is a relativized world in which a relatively natural complexity class-namely a downward closure of NP, \rsnnp - has…
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the…